Having identified the space of stability conditions \(\operatorname{Stab}^\dagger \mathcal {C}\)  inside the two-dimensional space of central charges parametrized by \((T,T_D)\) , it remains to determine which part of it is covered by the physical slice of \(\Pi \) -stability conditions. As mentioned in the introduction and further detailed in Appendix , the slice \(\Pi \)  is isomorphic to the universal cover of the modular curve \(X_1(3)\) , and conveniently parametrized by \(\tau \)  in the Poincaré upper half-plane \({H}\) .
As shown in
[1]}, there is an embedding \({H}\hookrightarrow \operatorname{Stab}^\dagger \mathcal {C}\)  sending
\(\tau \mapsto (Z_\tau ,\mathcal {A}_\tau )\)  which is equivariant with respect to the action of \(\Gamma _1(3)\)  on both sides. The central charge function \(Z_\tau \)  (REF ) is determined by the
representation (REF ) for the periods,
\(T(\tau )= -\frac{1}{2} + \int _{\tau _o}^{\tau } C(u) \mathrm {d}u\ ,\quad T_D(\tau ) = \frac{1}{3} + \int _{\tau _o}^{\tau } C(u) \,u \mathrm {d}u\) 
